Also, the ratio of the lengths of each new curve to the old curve is: <math>\frac{4(s\sqrt{2})}{4(s)} = \sqrt{2} </math>, so that <math>e = \sqrt{2}</math>.

Also, the ratio of the lengths of each new curve to the old curve is: <math>\frac{4(s\sqrt{2})}{4(s)} = \sqrt{2} </math>, so that <math>e = \sqrt{2}</math>.

-

Thus, the fractal dimension is <math>\frac{logN}{loge} = \frac{log2}{log\sqrt{2}} = 2 </math>, and it is a <balloon title="A space-filling curve in 2-dimensions is a curve with a fractal dimension of exactly 2. This means that the curve touches every point in the unit square.>space-filling curve</balloon>.

+

Thus, the fractal dimension is <math>\frac{logN}{loge} = \frac{log2}{log\sqrt{2}} = 2 </math>, and it is a <balloon title="A space-filling curve in 2-dimensions is a curve with a fractal dimension of exactly 2. This means that the curve touches every point in the unit square."> space-filling curve</balloon>.

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}}

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Revision as of 15:27, 1 July 2009

This image is an artistic rendering of the Harter-Heighway Curve (also called the Dragon Curve), which is a fractal. This curve is an iterated function system and is often referred to as the Jurassic Park Curve, because it garnered popularity after being drawn and alluded to in the novel Jurassic Park by Michael Crichton (1990).

Basic Description

This fractal is described by a curve that undergoes a repetitive process (called an iterated process). To begin the process, the curve has a basic segment of a straight line.

Then at each iteration:

Each line is replaced with two line segments at an angle of 90 degrees (other angles can be used to make fractals that look slightly different)

Each line is rotated alternatively to the left or to the right of the line it is replacing

Base Segment and First 5 iterations of the Harter-Heighway Curve

15th iteration

The Harter-Heighway Dragon is created by iteration of the curve process described above. This process can be repeated infinitely, and the perimeter or length of the dragon is in fact infinite. However, if you look to the image at the right, a 15th iteration of the Harter-Heighway Dragon is already enough to create an impressive fractal.

An interesting property of this curve is that the curve never crosses itself. Although the corners of the fractal seem to touch at various points, the curve never actually crosses over itself. Also, the curve exhibits self-similarity when iterated infinitely, because as you look closer and closer at the curve, the curve continues to look like the larger curve.

A More Mathematical Explanation

Note: understanding of this explanation requires: *Algebra

Properties

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Perimeter

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1st iteration of the Harter-Heighway Dragon

The perimeter of the Harter-Heighway curve increases by a factor of for each iteration.
For example, if you look at the picture to the right, the straight red line shows the fractal as its base segment and the black crooked line shows the fractal at its first iteration.

If the first iteration is split up into two isosceles triangles, the ratio of the first iteration over the base segment is:

Number of Sides

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The number of sides () of the Harter-Heighway curve for any degree of iteration (k) is given by , where the "sides" of the curve refer to alternating slanted lines of the fractal.

For example, the third iteration of this curve should have a total number of sides .

Fractal Dimension

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The Fractal Dimension of the Harter-Heighway Curve can also be calculated using the equation: .

Let us use the second iteration of the curve as seen below to calculate the fractal dimension.

There are two new curves that arise during the iteration so that .

Also, the ratio of the lengths of each new curve to the old curve is: , so that .

Thus, the fractal dimension is , and it is a space-filling curve.

Changing the Angle

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The Harter-Heighway curve iterates with a 90 degree angle. However, if the angle is changed, new curves can be created: